There appears to be conflicting experimental evidence on the redistribution of the electron density in the lone-pair and other regions of a molecule due to the interaction with its nearest neighbours. In some experimental as well as theoretical deformation density maps a decrease in the lone-pair density has been reported, whereas in other cases an increase has been found. It appears that two major, counteracting factors are responsible for these differences (apart from experimental errors in the diffraction studies and limited accuracy in the theoretical calculations): an increase in the lone-pair density is expected due to the polarizing influence of the neighbours, whereas simple superposition of the isolated monomer deformation densities will lead to an apparent decrease due to the overlap with the negative contours of the neighbouring atom. Depending on which of these factors is the dominant one, an increase or decrease may thus be observed. These points are illustrated by recent results on nickel sulfate hexahydrate and some other hydrogen-bonded compounds. The electron density based on the fitted deformation functions of all atoms in the structure is compared with the individual densities calculated from deformation functions of the separate monomers. In this way the effects of simple superposition of the individual densities have been studied, and a partitioning of the electrostatic and polarization contributions to the hydrogen bonds and other relatively weak bonds to the oxygen lone-pairs attempted.

X-ray diffraction data in heavy-atom compounds may be sensitive to anharmonic atomic displacements, since the large core electron densities result in appreciable scattering amplitudes at large reciprocal distances. Since bonding electron densities may also exhibit sharp features affecting high-order reflections, they may be difficult to distinguish from anharmonic effects. We have ana-lyzed an accurate room-temperature single-crystal X-ray data set of K 2 PtCl 6 using least-squares anharmonic displacement and charge density formalisms. The Hirshfeld charge density formalism, which has successfully been applied to many light-atom structures, fails to parametrize satisfactorily the data, whereas the electron densities at K and CI are easily accounted for by an anharmonic Gram-Charlier expansion to 4th order. Densities around Pt are parametrized only by a combination of anharmonicity and charge density formalisms. If economical parametrizations of the experimental data are preferred to more complicated ones, anharmonicity may be conjectured to play an impor-tant role while the main bonding feature consists of a preferential occupation of the 5d-orbitals of Pt with t 2g symmetry.

The effect of anharmonic atomic vibrations on the elastic scattering of X-rays by a crystal is treated by time-independent perturbation theory. The complete expression for the anharmonic temperature factor (TF) in the high-temperature limit is presented for atoms in general positions. The phonon-phonon interactions in an anharmonic crystal may be identified with X-ray diffraction data by using various temperature and scattering vector q dependencies of different anharmonic contri-butions to the TF. The expression for the intensity of elastic scattering by an anharmonic crystal contains an additional factor -the anharmonic cross-term. This term has to be taken into account for obtaining the unbiased thermal parameters of fourth order in | q | from X-ray diffraction data.

Owing to the close neighbourhood of Ga and As in Mendeleev's table, GaAs shows two funda-mental classes of X-ray structure amplitudes distinguished by their extremely different scattering power. They are differently sensitive to the valence electron density (VED) redistribution caused by the chemical bond and must be measured by different experimental methods. Using such data, both the VED and the difference electron densities (DED) are calculated here. Comparison with theoret-ical densities shows that the VED is characterized by covalent, ionic and metallic contributions. The DED constructed from GaAs and Ge data demonstrates the electronic response caused by a "protonic" charge transfer between both f.c.c. sublattices as well as the transition from a purely covalent to a mixed covalent-ionic bond. Especially the charge-density accumulation between nearest neighbours (bond charge (BC)) depends on the distance between the bonding atoms and changes under the influence of any lattice deformation. This phenomenon is described by a BC-trans-fer model. Its direct experimental proof is given by measuring the variation of the scattering power of weak reflections under the influence of an external electric field. This experiment demonstrates that the ionicity of the bond changes in addition to the BC variation.

The elctron density distribution in crystalline ZnSe and ZnTe has been measured by X-ray diffraction on powder samples and is compared with the results for other zincblende-type crystals. It is shown that the data for these two compounds corroborate the linear dependence on the bond charge that has been found earlier for the effective atomic charge from infrared reflection measure-ments, the optical dielectric constant and the ionicity estimated from pseudopotential band theory.

The electron density and bonding in the silicates beryl, cordierite and dioptase has been investi-gated. The replacement of Be, Si and Al atoms in the identical structural fragments leads to a (^-redistribution and changes the cr-bond character from slightly polar to strongly polar. The condensation of [SiOJ-tetrahedra in the rings leads to the accumulation of electron density near bridge oxygens. This phenomenon is expressed especially in dioptase (pure ring silicate). Therefore, the electron density picture may be considered as a supplementary independent criterion for classi-fication of crystal structures. The experimental (5@-maps for H 2 0 molecules in dioptase are in good agreement with theoretical ones. The ^^-distribution of 3 d-electrons and bonding in the Jahn-Teller distorted Cu-octahedron can be interpreted from crystal-field-theory point of view.

X-ray diffraction intensities from a single crystal of the Laves phase FeBe 2 were collected at room temperature. Atomic and thermal parameters as well as a scale factor were determined by least-squares refinements of high-angle data [(sin d)/X 0.6Ä -1 ]. The results of Fourier inversion of the data are presented in terms of maps of the total density and followed by a discussion of the "effects" of two criteria for labelling reflection intensities as "unobserved", based on their statistical accuracy.

The low-temperature molecular structure and electron density (ED) distribution of dioxane and trans-2,5-dichloro-l,4-dioxane were determined near 100 K from X-ray data. Deformation electron density maps were calculated by Fourier and multipole expansion techniques. The multipole model charge distribution of dioxane determined from experimental data was taken as a reference state, relative to which the ED of the substituted ring was analyzed. These fragment deformation density maps show quadrupolar deformations on atoms O, CI, and anomeric C only. The highly correlated orientation of the lobes at these atomic sites seems to indicate that the observed charge rearrange-ment is caused by three-center interactions.

A 20 K difTractometer for X-ray measurement is described. Electron density experiments at 15 K for oxalic acid and at 23 K for acetamide were carried through, leading to high-resolution experi-mental and multipole static density maps.

One-bond, 13 C-X H coupling constants, Jj(C-H), in amines, ammonium ions, and carboxylic amides correlate with structure and support the concept that the value of J^C-H) is related to the charge density on the nitrogen atom; for example, amine oxides have nearly the same charge density at nitrogen as does the tetramethylammonium ion. The Jj(C-H) values for methyls bonded to nitrogen in various amides then give an experimental estimate of the charge density at the nitrogen atom that enables an estimate of the bond order in the C-N amide-bond; the data suggest that carboxylic amides have a C-N bond order of about 1.35, that sulfonamides have an S-N bond order of about 1.45, and that phosphinamides, R 2 P(0)N(CH 3) 2 , have a P-N bond order of about 1.3. In contrast, aminephosphines have a P-N single bond. The value for carboxylic amides is in reasonable agreement with bond distances in amides.

The requirements for charge density studies of more complex molecules have been realized by use of the FAST area-detector diffractometer in conjunction with a rotating-anode generator, which allows to collect large quantities of diffraction data within less than two days, independent of unit cell size. Oxalic acid dihydrate has been chosen for test experiments. Data collection strategies are described and preliminary results are presented. These show that data of sufficiently high quality for charge density studies can be collected.

Incomplete and imperfect data characterize the problem of constructing electron density represen-tations from experimental information. One fundamental concern is identification of the proper protocol for including new information at any stage of a density reconstruction. An axiomatic approach developed in other fields specifies entropy maximization as the desired protocol. In particular, if new data are used to modify a prior charge density distribution without adding extraneous prejudice, the new distribution must both agree with all the data, new and old, and be a function of maximum relative entropy. The functional form of relative entropy is a = — g In (g/z), where g and x respectively refer to new and prior distributions normalized to a common scale. Entropy maximization has been used to deal with certain aspects of the phase problem of X-ray diffraction. Varying degrees of success have marked the work which may be roughly assigned to categories as direct methods, data reduction and analysis, and image enhancement. Much of the work has been expressed in probabilistic language, although image enhancement has been somewhat more physical or geometric in description. Whatever the language, entropy maximization is a specific and deterministic functional manipulation. A recent advance has been the description of an al-gorithm which, quite deterministically, adjusts a prior positive charge density distribution to agree exactly with a specified subset of structure-factor moduli by a constrained entropy maximization. Entropy on an iV-representable one-particle density matrix is well defined. The entropy is the expected form, and it is a simple function of the one-matrix eigenvalues which all must be non-neg-ative. Relationships between the entropy functional and certain properties of a one-matrix are discussed, as well as a conjecture concerning the physical interpretation of entropy. Throughout this work reference is made to informational entropy, not the entropy of thermodynamics.

The nature of the bonding in Be metal was studied by investigating the MEM map, which is the electron density distribution obtained by the Maximum-Entropy Method. In order to avoid extinc-tion effects, 19 Bragg reflections were measured by a new powder-diffraction experiment that utilizes Synchrotron Radiation as an incident X-ray and an Imaging Plate as detector. The experiment was carried out at the Photon Factory BL6A2. In spite of the limited number of reflections used in the MEM analysis, the electron density distribution of Be was obtained accurately and reliably. The structure factors for unmeasured reflections were calculated and compared with the values observed by Larsen and Hansen [Acta Cryst. B40, 169 (1984)]. The agreement is very good. Furthermore, the MEM map revealed that Be metal forms an electronic layer in the shape of a honeycomb that is parallel to the basal plane.

A procedure for calculating the electrostatic potential and the electrostatic energy of an ion in a crystal is presented. It is based on a mixed direct and reciprocal space approach, and it takes into account the detailed charge density distribution in the crystal which can be obtained from accurate X-ray diffraction measurements.

It has been shown (Z. Su and P. Coppens, Acta Cryst. A 48, 188 (1992)) that the electrostatic potential, the electric field, and the electric field gradient (EFG) can be expressed in closed forms in terms of the positions and the charge-density parameters of individual atoms, whose aspherical density is described by a pseudoatom model (e.g., N. Hansen and P. Coppens, Acta Cryst. A 34, 909 (1978)). A Fortran program Molprop91 based on this method has been written (Z. Su, State Univ. of New York at Buffalo 1991). The method has been applied to the title compounds. Low-temper-ature X-ray diffraction data of fully deuterated benzene (G. A. and d,/-histidine (N. Li, Ph.D. thesis, State University of New York at Buffalo 1989) were analyzed using the least-squares deformation density refinement program Lsmol90 (a modified version of Molly). Molprop91 was subsequently used to calculate the electrostatic-poten-tial maps in selected sections, and at the nuclear positions. For the latter, the EFGs were also evaluated. The electrostatic potentials were used to fit net atomic charges and estimate the molecular energies. Errors in the derived quantities are given.

The calculation of the electrostatic potential of a molecule removed from the crystal lattice is derived from the parameters obtained by a kappa refinement and by a Hansen-Coppens electron density model. These calculations in direct space are applied to N-acetyl-a,/?-dehydrophenylalanine; deformation potentials calculated by Fourier transformation are compared to those obtained in direct space.

A total of about 37 000 diffracted intensities has been measured at 20 K for a spherical single crystal of citrinin. Using a multipole formalism to interpret the X-ray data, maps of the charge density and of its Laplacian, as well as for the electrostatic potential have been derived. A value of 7(2) D has been obtained for the magnitude of the molecular dipole moment. A study of the electric field gradient (EFG) at the nuclei has yielded the atomic quadrupole coupling constants (QCC) and asymmetry parameters (r\). A topological analysis of the charge density has been performed to characterize the intramolecular covalent and hydrogen bonds. tures, in the sequence: 293, 20, 240, and again 293 K. From the combined X-ray results of the two investiga-tions, the thermodynamic parameters for the proton transfer occurring in the crystals of citrinin have been derived [2]. Furthermore, it has been shown that it can be safely assumed that the 20 K X-ray structure of citrinin describes the pure p-quinonemethide form. In h16b 4a 018 Fig. 1. Atomic numbering scheme for citrinin.

The bonding type and characteristic chemical reactions of the sydnone ring have been rationalized by examination of the deformation electron density distribution and electrostatic potential from both experiment and theory. The net atomic charges, bond lengths and bond orders of the title compound confirm the semiaromatic bonding type for the sydnone ring. The highly negative net atomic charge of C4 suggests its susceptibility to electrophilic substitution at this position. The well known 1,3-dipolar cycloaddition reaction of the sydnone ring can be understood by the analysis of the 7 t -o r b i t a l wavefunctions, which give rise to similar deformation densities as the experimental ones. The good agreement between the experimental and theoretical (ab initio calculation using 3-21G basis sets) deformation density distribution of the title compound justifies the use of the same basis set to derive the electrostatic potential. The electrostatic potential map reveals all the possible protonation sites of the sydnone ring at Ol, N2 and 06 with the deepest hole at 06 (84.6 kcal/mole). The molecular structure of 3-(p-ethoxyphenyl)syd-none [1] is shown in Figure 1. The phenyl ring plane and the sydnone ring plane are nearly coplanar (di-hedral angle of 2°). A deformation density study of this compound is presented here using both experimental X-ray diffraction data at 110 K and a split valence basis ab-initio calculation. The deformation densities examined are Ag x _ x , Ag m _ a , and Ag mo _ ao , where ÄQx-x is from conventional X-X (high angle [2], Ag m _ a is from a multipole model [3], Ag mo _ ao is from an ab-initio calculation using a 3-21G basis set [4]. The bond lengths, bond orders and net atomic charges of this compound are presented in Table 1 with com-parison between experimental and theoretical results. Bond lengths and bond orders indicate C5-06 to be Fig. 1. Molecular drawing with thermal ellipsoids at 110 K.